Heading reference command and control algorithm systems and methods for aircraft turn-to-target maneuvers

ABSTRACT

Systems and methods are provided for determining a final heading of a turning vehicle, such as a rotorcraft. The system may include an algorithm that calculates an advance prediction of a final heading that will be achieved after control input is terminated.

This invention was made with Government support under contract numberDAAH 10-00-C-0052 awarded by the United States Army. The Government hascertain rights in this invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is related to vehicle control, and more specifically, tosystems and methods for precisely and quickly pointing a vehicle in adesired direction.

2. Description of the Related Art

Turning a machine, such as a vehicle, or a portion thereof, quickly topoint in a desired direction can be difficult. Turning precisely andquickly can be a challenge for flying and/or hovering vehicles, such as,for example, a helicopter or other rotorcraft (e.g., a tilt-rotoraircraft, a vertical/short takeoff and landing (VSTOL) aircraft),whether it is piloted or an unmanned aerial vehicle (UAV). Typically,internal and external effects, such as moments of inertia, angularmomentum, structural limits, aerodynamic loads, and control surfaceactuator authority and rate limits must be dealt with.

For military aircraft, it may be especially important to have acapability to point a vehicle quickly in a desired direction, forexample, to acquire, track, and fire upon an enemy target in advance ofthe enemy firing upon the aircraft. In this regard, the United StatesArmy has established a performance specification entitled “AERONAUTICALDESIGN STANDARD (ADS-33E-PRF) HANDLING QUALITIES REQUIREMENTS FORMILITARY ROTORCRAFT,” that includes the following requirement for flightin Good Visual Environment (GVE) conditions:

“3.11.17 Turn to Target (Handling Qualities Requirement)

From a stable hover at an altitude of less than 20 ft complete a 180 degturn. Turns must be completed in both directions. Final rotorcraftheading must be achieved within 5 seconds of initiating the turn within+/−3 degrees of the target.”

This task may be difficult with state-of-the-art technology because itis typically difficult for a pilot to time his or her directionalcontrol inputs precisely enough to get the aircraft to stop at theintended terminal heading without significant heading overshoot and timeconsuming heading backup toward the target heading. The headingovershoot and resulting heading backup problems are a result of aircraftyaw acceleration and jerk (time derivative of acceleration) limitationsthat prevent the pilot from instantaneously arresting the yaw ratedeveloped during the turn.

In single rotor helicopters, these yaw acceleration and jerk limitationsare imposed by the combined effect of tail rotor gearbox torque limits,tailrotor collective pitch actuator authority and rate limits, andlimitations on allowable yaw acceleration due to inertial structuralload limitations on attachment points for external stores.

Tandem rotor helicopters and tiltrotor aircraft face similar limits onallowable yaw acceleration and jerk due to rotor flapping induced bladeand hub loads and differential cyclic pitch actuator authority and ratelimits. Thus, the rapid turn to target task is difficult in virtuallyall vehicles with Vertical Takeoff Or Landing (VTOL) capabilities.

Meeting this requirement with existing rotorcraft control systems andconfigurations would rely heavily on pilot skill and training. Forexample, a pilot would need to anticipate, while turning a rotorcraft,the heading (azimuth) that the rotorcraft would achieve at the end ofthe turn once input force is released from the controls (e.g., rudderpedals), largely relying on experience and “feel” of the rotorcraft'shandling characteristics, as well as mentally factoring in environmentalconditions such as wind, altitude, etc. This results in an undesirablyhigh pilot workload, and meeting the requirement using existingrotorcraft control systems and configurations could be impossible, orvery difficult. In addition, there is a possibility of a pilotinadvertently exceeding the mechanical limits of a rotorcraft tail rotorgearbox and/or airframe while attempting to turn to a target quickly.

One possible solution is to build a rotorcraft with a large tail rotor,such as was proposed to be included in the Comanche helicopter program.However, such a solution may require an excessive cost, as well asdesign tradeoffs, such as increased weight and reduced agility.

This disclosure is directed toward overcoming one or more problems ordisadvantages associated with the prior art.

SUMMARY OF THE INVENTION

According to one aspect of the invention, a computerized system predictsa final attitude of a maneuvering aircraft. The system includes acomputer processor and an algorithm programmed into the computerprocessor. The algorithm is adapted to calculate, during a maneuver, apredicted final aircraft attitude based on aircraft parameters. Thealgorithm may include calculations based upon model following controllaws.

According to another aspect of the invention, the algorithm may furtherinclude a plant canceller element. The final attitude may include afinal heading and/or a final pitch attitude.

In accordance with a further aspect of the invention, the modelfollowing control laws may be tailored to maintain aircraft angularacceleration within specified limits, such as, for example, limits ofangular acceleration and/or angular jerk about an aircraft yaw axis.

According to other aspects of the invention, the predicted finalaircraft attitude is calculated using one of an exact solutionalgorithm, an iterative solution algorithm, and an approximate solutionalgorithm.

In accordance with a still further aspect of the invention, a method ofpredicting a final attitude of a maneuvering aircraft is provided. Themethod includes providing a computer processor, programming an algorithminto the computer processor to calculate a predicted final aircraftattitude based on aircraft parameters, and calculating, during amaneuver, the predicted final aircraft attitude.

The features, functions, and advantages can be achieved independently invarious embodiments of the present invention or may be combined in yetother embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is diagrammatic plan view of a rotorcraft that may incorporatethe invention, shown in three phases of an exemplary turning maneuver;

FIG. 2 is a block diagram illustrating a system and method of turning anaircraft according to one aspect of the invention;

FIG. 3 is a block diagram illustrating a generic architecture that maybe used for explicit model following control laws;

FIG. 4 is a block diagram illustrating the architecture of FIG. 3 infurther detail;

FIG. 5 is a block diagram of a system and method that includes a yawaxis explicit model following control law architecture as modified toperform yaw acceleration and jerk limiting;

FIG. 6 is a plot that shows a time history of a final heading predictioncue curve overlaying a time history of commanded heading curve for a 180degree turn of an aircraft implementing the invention;

FIG. 7 is an illustration of an exemplary cockpit display that includesa final heading prediction icon according to the invention;

FIG. 8 is a plot that shows piloted simulation predictions for amaneuver in which data are presented for the maneuver conducted with andwithout using the invention;

FIG. 9 is bar graph that shows an average agility factor for threedifferently configured aircraft.

DETAILED DESCRIPTION

FIG. 1 illustrates the operation of a Heading Reference Command andControl Algorithm according to one aspect of the invention. Withreference to FIG. 1, a pilot 30 is faced with an example of a maneuverin which he or she has to execute a maximum rate hover turn ofapproximately 90 degrees to point a helicopter 32 at an intended target34, for example, a tank, within weapons firing constraints. The maneuvermay be broken down into three phases in FIG. 1: a first phase, generallyindicated at 36 in which the pilot 30 turns the helicopter 32 at amaximum yaw rate toward the target 34 in order to place a cursor 38displayed along with the target 34 on a visual display device such as ahelmet-mounted monocle 40; a second phase, generally indicated at 42, inwhich the pilot 30 relaxes directional control input force when thecursor 38 is displayed on the monocle 40 over the target 34; and a thirdphase, generally indicated at 44, in which the helicopter 32automatically aligns with a heading pointing toward the target 34,within firing constraints.

A Heading Reference Command & Control Algorithm, described in furtherdetail below, makes the rapid turn to target task illustrated in FIG. 1possible with a reasonable level of pilot workload by providing a“Heading Reference Cue” in the pilot's monocle 40. The monocle 40 may bean Integrated Helmet and Display Sighting System (IHADSS) monocle. TheIHADSS monocle-may be part of a synthetic vision system, and may projecta synthetic image of the outside world, such as, for example, an imagegenerated by a Forward Looking Infrared (FLIR) sensor (not shown). TheFLIR image may be a virtual image of the outside world that is enhancedby the FLIR sensor to provide improved vision at night or in otherdegraded visual environments.

The field of view of the IHADSS monocle may be slaved, or head-tracked,to the pilot's head, thus the pilot 30 may simply turn his or her headto view objects to the left and right of the helicopter 32 through theenhanced vision capabilities of the FLIR sensor. Flight displaysymbology may be superimposed over the FLIR image in the IHADSS monocle,allowing the pilot 30 to monitor critical flight display parameterswhile his or her vision is directed outside the cockpit.

The cursor 38 may be a “Heading Reference Cue,” that indicates theclosest possible heading at which the helicopter 32 can be stopped withno heading overshoot. The “Heading Reference Cue” may be fixed withrespect to the virtual image of the outside world depicted by the FLIRimagery in the IHADSS monocle.

When the pilot 30 uses the “Heading Reference Cue,” he or she may applyfull directional control input and easily command maximum yaw rateduring the turn to minimize a time line to a firing solution. As shownin FIG. 1, as the helicopter 32 turns toward the target 34, the “HeadingReference Cue” may move toward the target 34 in the pilot's IHADSSdisplay. When the “Heading Reference Cue” is superimposed over the imageof the target 34, the pilot 30 may simply relax directional controlinput force to command the helicopter 32 to stop at the target headingwith no heading overshoot. High fidelity simulations conducted under theHelicopter Active Control Technology (HACT) program predict that theHeading Reference Command & Control Algorithm will decrease time linesto target engagement in the scenario depicted by FIG. 1 by as much as60% in critical nighttime operational scenarios or in other degradedvisual environments.

While the preceding discussion focuses on providing a reference cue fora directional axis targeting task, the concept of providing an attitudereference cue and integrated control laws to lower pilot workload andshorten timelines required to achieve the target attitude may also beapplied to the other axes of control. For example, there are missionrequirements for pitch axis pointing tasks and limitations on aircraftpitch axis response capabilities that are analogous to those discussedin association with the directional axis task depicted in FIG. 1.

It is straightforward to extend the functionality of the cue depicted inFIG. 1 to serve as an attitude reference for both yaw and pitch axispointing tasks. For example, the cue in FIG. 1 may move horizontally andvertically with respect to the virtual FLIR image to cue the pilot toboth directional and longitudinal axis pointing requirements. When theheading/pitch attitude reference cue is superimposed over the intendedtarget in the FLIR image, the pilot could simultaneously relax inputforces in both the longitudinal and directional axis controls to bringthe aircraft to the required final heading and pitch attitudes with noovershoot and in the minimum amount of time.

It should be noted that the symbology for the Heading Reference Cuecould take many different forms than the symbol shown in FIG. 1. Thespecific symbol used to depict the Heading Reference Cue in FIG. 1 ispresented solely for the purpose of illustrating the concept through asingle simplified example. The specific size, shape, color, and geometryof the cue would vary depending upon the specific application.Similarly, multiple sources besides the line of sight from the pilot'seye to the target depicted in FIG. 1 could be used to drive the HeadingReference Cue. For example, lines of sight from Target Acquisition andDesignation System (TADS) sensors, which may be located at variouspoints on the aircraft, may be used for targeting tasks.

It should also be noted that the Heading Reference Command and ControlAlgorithm is equally applicable to aircraft equipped with conventionalcenter stick and pedal cockpit controls or aircraft equipped withsidestick controllers. In an aircraft equipped with conventionaldirectional pedals, the pilot may relax directional control input forceby relaxing his leg muscles and allowing the directional pedals tocenter. In an aircraft equipped with a twist grip sidestick controllerto control the yaw axis, the pilot may relax directional control inputforce by relaxing his wrist muscles and allowing the torsional axis ofthe sidestick controller to center. The pilot may relax directionalcontrol input force at an appropriate time indicated by the cue tocommand the aircraft to stop at the target heading with equivalentproficiency whether the aircraft is equipped with directional pedals orsidestick controllers.

FIG. 2 is a block diagram illustrating one example of system leveloperation of the Heading Reference Command & Control Algorithm. Anoverall system 46 may include an Aircraft Response Element 48, a ModelFollowing Control Laws Module 50, Final Heading Prediction AlgorithmModule 52, a Final Heading Cue Display Module 54, and/or a Pilotresponse element 56. Referring to FIG. 2, an input to the overall systemmay be a “Desired Final Heading” and an output of the overall system maybe an actual aircraft heading. The Heading Reference Command & ControlAlgorithm effectively minimizes the error between the actual aircraftheading and the desired final heading by providing the pilot with adisplay cue that allows the pilot to close the loop on the final headingwith a low workload and minimal training.

The Aircraft Response Element 48 represents the basic heading and yawrate response of the aircraft to a directional actuator command input(δ_(RACT)). For most VTOL aircraft, the basic heading response of theaircraft to actuator commands is too sluggish to provide precisepointing capabilities with a reasonable level of pilot workload. TheModel Following Control Laws Module 50 may shape the pilot'sdirectional-control inputs to provide crisp heading and yaw rateresponse characteristics that lower pilot workload required to controlthe directional axis. The Model Following Control Laws Module 50 mayalso provide feedback of sensed aircraft heading and yaw rate fordisturbance rejection and stability enhancement. The Model FollowingControl Laws Module 50 may replace the sluggish inherent dynamics of theaircraft, defined approximately by the bare airframe yaw rate damping(N_(R)) and yaw acceleration control sensitivity (N_(δR)) derivatives,with the desired crisp and predictable yaw axis handling qualities of afirst order system with the desired bandwidth (K₁) and maximumachievable yaw rate specified by military handling qualitiesspecifications such as Aeronautical Design Standard (ADS) 33.

The Heading Reference Command & Control Algorithm system 46 advances thestate of the art by introducing aircraft structural limits on yawacceleration and jerk into the model following control laws. By limitingactual aircraft yaw acceleration and jerk to the values used as inputsto the Final Heading Prediction Algorithm Module 52, the Model FollowingControl Laws Module 50 helps to ensure accuracy of the Final HeadingPrediction Algorithm Module 52 by rejecting errors between actual andassumed maximum yaw acceleration and jerk capabilities, for example, dueto unmodelled aircraft dynamics or atmospheric turbulence. In additionto improving the accuracy of the Final Heading Prediction AlgorithmModule 52, the Model Following Control Laws Module 50 may also activelycontrol the helicopter 32 to ensure that the desired final heading isattained with no residual heading error or overshoot when the pilot 30reacts to the display cue 38 converging with the target 34 by relaxingdirectional control input force.

While the block diagram shown in FIG. 2 is drawn for the yaw axis, asimilar block diagram could be drawn for the pitch axis. The input tothe pitch axis block diagram analogous to FIG. 2 would be “Desired FinalPitch Attitude” and its output would be “Aircraft Pitch Attitude.”Referring to FIG. 2, the “Final Heading Prediction Algorithm” Module 52may be replaced by a “Final Pitch Attitude Prediction Algorithm” Module,and the “Final Heading Cue Display” Module 54 may be replaced by a“Final Pitch Attitude Cue Display” Module.

FIG. 3 shows a generic architecture that maybe used for explicit modelfollowing control laws. The generic model following control lawarchitecture provides an aircraft response to pilot commands that isequal to a desired response command model “D(s)” 58. A Plant CancellerModule 60 cancels the inherent aircraft dynamics, denoted by P(s) inFIG. 3, while substituting the desired response of the command model,denoted by D(s) in FIG. 3. Any errors between the desired responsecommand model and sensed aircraft response data may be passed through aStability Compensation Module 62 and fedback through a Limited AuthorityPort 64 to actively mitigate model following errors. A Sensor Module 66may be used to provide the sensed aircraft response data.

Errors between the command model response and actual sensed aircraftresponse, referred to as model following errors, may be caused byexternal disturbances such as wind gusts as well as inexact cancellationof the aircraft dynamics by the Plant Canceller Module 60. Since theexplicit model following control law architecture minimizes errorsbetween sensed and commanded aircraft response, the output of theStability Compensation Module 62 may be limited to a relatively smallmagnitude by the limited authority port without affecting performance ofthe system. Limiting the port authority provides robustness to sensorfailures by allowing the pilot enough authority to easily override anyerroneous stability compensation inputs caused by sensor failures. Theaircraft module 68 represents the system being controlled by the modelfollowing control laws, but of course the system being controlled couldbe another type of machine, such as a robot arm or a crane, that mayneed to be precisely pointed in a desired direction.

FIG. 4 shows how the generic explicit model following control lawarchitecture may be applied to the yaw axis of the aircraft.

-   D(s)=Transfer Function of Desired Response Command Model-   H(s)=Transfer Function of Control Law Stability Compensation Element-   K_(r)=Yaw Rate Error Feedback Gain (in/(rad/sec))-   K_(Ψ)=Heading Error Feedback Gain (in/rad)-   N_(R)=Yaw rate damping derivative (rad/sec)-   N_(δR)=Yaw rate control response derivative (rad/sec    2/in)-   P(s)=Transfer Function of Unaugmented Aircraft Response-   s=Laplace Variable (rad/sec)-   δ_(R)=Directional Control Input (%)-   {dot over (Ψ)}_(MAX)=Maximum Yaw Rate (Units=Radians/sec)

The input may be a pilot directional control input in units of percentand the output may be aircraft heading. A small deadzone function 70, onthe order of +/−5% of full control authority, may be applied to thepilot directional control input so that the pilot can easily commandzero yaw rate by relaxing directional controller input force even if thecontroller does not return to an exactly centered position. A nonlinearshaping function 72 may be applied to the output of the deadzone so thatcontrol sensitivity is lower for small control inputs and higher forlarger control inputs. The nonlinear shaping provides precise controlfor small heading changes while allowing the pilot to exploit full yawrate capability for large control deflections. The output of thenonlinear shaping may be divided by a factor of 100% and multiplied bythe maximum allowable yaw rate to scale the yaw rate command so that afull authority directional control input commands the maximum allowableyaw rate.

The Plant Canceller Module 60 may be designed based on the assumptionthat the aircraft directional axis dynamics can be represented by afirst order response characterized by yaw rate damping and directionalcontrol response sensitivity derivatives. The first order responseassumed for the aircraft dynamics is an approximation. For example,additional unmodelled dynamics may be present in the aircraft responseand the yaw rate damping and directional control response sensitivityderivatives may differ from the assumed values as gross weight, centerof gravity, and atmospheric conditions vary.

The explicit model following control law architecture is robust toreasonable discrepancies in assumed and actual aircraft dynamics,therefore adequate plant cancellation is achieved as long as the assumedaircraft dynamics roughly match the actual aircraft dynamics. Ingeneral, adequate model following performance is obtained as long as theyaw rate damping and directional control response sensitivityderivatives assumed in the aircraft model are within +/−50% of thevalues that describe an equivalent first order response model of theactual aircraft dynamics.

As shown in FIG. 4, the Plant Canceller Module 60 may be a function ofdesired yaw axis control response bandwidth, bare airframe yaw damping,and bare airframe directional control response sensitivity. The yaw ratedesired response command model 58 may be formed by cascading a firstorder lag filter with the desired yaw axis control response bandwidth(K₁) with models of actuator dynamics and system digital and transportdelays. Commanded yaw rate may be integrated to calculate the commandedheading angle. Yaw rate error may be calculated by subtracting sensedyaw rate from commanded yaw rate. Heading error may be calculated bysubtracting sensed heading angle from commanded heading angle. Stabilitycompensation may be calculated by multiplying the yaw rate and headingerrors by appropriate feedback gains and structural filters. Thestability compensation command may be limited by the port before beingfedback to the directional control actuator to provide robustness tosensor failures.

The generic explicit model following control law architecture may alsobe applied to the pitch axis of the aircraft using the same or similararchitecture shown in FIG. 4. Pitch rate damping, pitch control responsesensitivity, pitch axis desired bandwidth, maximum pitch rate, pitchattitude feedback gain, and pitch rate feedback gain would replace theiryaw axis counterparts in the explicit model following control lawarchitecture block diagram shown in FIG. 4.

FIG. 5 is a block diagram of a system and method that includes a yawaxis explicit model following control law architecture as modified toperform yaw acceleration and jerk limiting. The block diagram of FIG. 5differs from that of FIG. 4 in that the dynamics of the Plant CancellerModule 60 and desired response command model 58 are combined into anintegrated yaw rate command model 74. It is possible to combine thePlant Canceller Module 60 and the desired response command model 58because the denominators of the transfer functions of both elementscontain the same poles. The integrated yaw rate command model calculatescommanded yaw rate as the output of an integrator, referred to as theyaw rate command integrator. Authority limiting the input to the yawrate command integrator limits commanded yaw acceleration, while ratelimiting the input to the yaw rate command integrator limits commandedyaw jerk. The explicit model following control law architecture shown inFIG. 5 provides equivalent model following performance as thearchitecture shown in FIG. 4, while adding the additional capability tolimit yaw acceleration and jerk precisely within explicitly specifiedvalues.

A pitch axis explicit model following control law architecture thatperforms acceleration and jerk limiting may also be implemented usingarchitecture analogous to that shown in FIG. 5 with the appropriatesubstitution of pitch axis parameters for the corresponding yaw axisparameters in FIG. 5.

FIG. 6 shows a time history of the final heading prediction cue curve 82(solid curve) overlaying a time history of commanded heading curve 84(dashed curve) for a case where the aircraft performs a 180 degree rightturn followed by a 180 degree left turn. The final heading predictioncue calculates the final heading associated with the release ofdirectional control input force. Referring to FIG. 6, at time equal to76 seconds the time rate of change of heading angle is virtually zero,thus the aircraft can obviously be stopped at the currently commandedheading. Therefore, at time equal to 76 seconds, the final headingprediction is identical to the currently commanded heading angle.Between time equal to 76 and 78 seconds, the time rate of change ofheading increases to a large value as the pilot conducts the right turn.

As the time rate of change of heading increases, the final headingprediction cue curve 82 diverges from the commanded heading curve 84,indicating that the final heading that will result from releasingdirectional control input force will be slewed significantly to theright of the currently commanded heading angle. For example, asillustrated in FIG. 6, at time roughly equal to 78 seconds, thecommanded heading is roughly 250 degrees while the predicted finalstopping heading is roughly 300 degrees. This means that if the pilotwere to release directional control input force at time equal to 78seconds that the aircraft would continue-to yaw an additional 50 degreesbefore it comes to a stop.

As the heading rate slows between time equal to 80 and 82 seconds, thefinal heading prediction cue curve 82 once again converges to thecommanded heading curve 84. The final heading prediction cue divergesagain from the commanded heading curve 84 as the aircraft initiates theleft turn between time equal to 82 and 84 seconds, indicating that thefinal stopping heading will be slewed significantly to the left of thecurrently commanded heading angle when the leftward time rate of changeof heading angle is large.

FIG. 7 shows a final heading prediction cue display 86 that can be usedas an alternative to the one illustrated in FIG. 1 in aircraft that arenot equipped with the synthetic visionics capability and/or head-upflight display capability of the Integrated Helmet and Display SightingSystem (IHADSS). The final heading prediction cue display shown in FIG.7 is a low cost option that indicates the predicted final stoppingheading by a diamond shaped symbol 88 displayed on a heading tape 90.The current heading angle is the heading shown in the center of theheading tape 90. The current heading angle may also be boxed to make iteasier for the pilot to recognize the current heading angle. FIG. 7depicts a high yaw rate right hover turn, wherein the current headingangle is 360 degrees and the predicted final stopping heading is 70degrees.

The simplified heading tape based final heading prediction cue displayshown in FIG. 7 can be used in aircraft equipped with night visiongoggles as well as aircraft equipped only with conventional head-downflight displays. High fidelity piloted simulations using the finalheading prediction cue display of FIG. 7 indicate that this simplifieddisplay provides significant benefit for demanding turn to targetmaneuvers such as those defined in Aeronautical Design Standard 33(ADS-33).

The specific size, shape, color, geometry, and location on the flightdisplay symbology of the cue may vary from that shown in FIG. 7depending upon the specific application. The specific symbol used todepict the Heading Reference Cue in FIG. 7 is presented solely for thepurpose of illustrating through a single simplified example how theconcept can be applied in aircraft without synthetic visionicscapabilities.

The U.S. Army developed the ADS-33E Turn to Target Mission Task Element(MTE) to measure directional axis handling qualities, maneuverability,and agility. The Turn to Target Mission Task Element (MTE) definesquantitative adequate and desired performance criteria for headingcontrol and position maintenance for an aggressive 180 degree headingturn. For Level 1 handling qualities in Attack helicopters, ADS-33Erequires that the 180 degree Turn to Target MTE be completed in lessthan 5.0 seconds in daytime conditions and less than 10 seconds innighttime conditions where the degraded visual environment makes theaircraft harder to fly. Given the current emphasis on nighttime tactics,it would obviously be desirable to provide equivalent agility in theTurn to Target MTE in both day and night visual conditions.

FIG. 8 shows piloted simulation predictions for the Turn to Target MTEconducted with an Apache helicopter in nighttime conditions. Data arepresented for the Turn to Target maneuver conducted with and without theFinal Heading Prediction cue generated by the Heading Reference Commandand Control Algorithm. As shown in FIG. 8, it takes 7.6 seconds for thepilot to complete the Turn to Target maneuver without the Final HeadingPrediction cue, as indicated by a solid curve 92. When the Final HeadingPrediction cue is provided, the pilot is able to reduce the time ittakes to perform the maneuver to 4.1 seconds, as indicated by a dottedcurve 94. Further, the Final Heading Prediction cue allows the maneuverto be conducted with no heading overshoot, providing an absolutelydeadbeat and predictable yaw rate response during the deceleration tothe target heading.

The time it takes to complete the Turn to Target maneuver is a measureof maneuverability and agility. Handling qualities rating is anotherdistinct metric that measures pilot workload required to achieve a givenlevel of maneuverability and agility. In other words, for a givenhelicopter and flight control system, it may be possible for a pilotwith exceptional skill to perform the Turn to Target MTE in 5 seconds,whereas a pilot with average skill may not be able to perform themaneuver in 5 seconds. Also, a pilot may be able to improve performanceby devoting more attention, or increasing workload, to perform the taskwithin the desired tolerance and allotted time. The Cooper Harper PointRating (CHPR) system provides quantitative handling qualities ratingsthat measure the ability of service pilots to perform Mission TaskElements (MTEs) such as the Turn to Target maneuver in operationalconditions. On the Cooper-Harper scale, lower numerical ratings indicatebetter handling qualities, with a CHPR of 1 indicating “best” handlingqualities and a CHPR of 10 indicating worst handling qualities.

Piloted simulation predicts that handling qualities ratings for the Turnto Target MTE improve from CHPR 7 for current fleet Apaches to CHPR 3when the current fleet Apaches are equipped with a modified flightcontrol system that includes the Heading Reference Command and ControlAlgorithm. A Cooper-Harper handling qualities rating of 7 indicates thatthe pilot could not perform the task to even adequate performancestandards with the current Apache flight control system even with a highlevel of pilot workload. A Cooper-Harper handling qualities rating of 3indicates that the modified flight control system allows the pilot toperform the task to desired performance standards with a low level ofpilot workload. Thus the Heading Reference Command and Control Algorithmsignificantly improves handling qualities while it increases usablemaneuverability and agility.

FIG. 9 is a bar graph that quantifies the progress toward achievement ofthe U.S. Department of Defense's (DoD's) Rotary Wing Vehicle TechnologyDevelopment Approach (RWV TDA) Subarea Goals for maneuverability andagility resulting from using flight control systems that incorporate theHeading Reference Command and Control Algorithm according to one aspectof the invention. The data shown in FIG. 9 quantify improvements inUsable Agility/Maneuverability that were measured within the context ofthe U.S. Army HACT Science and Technology program. The metric for thisSubarea Goal is average usable agility factor in aggressive maneuvers.Agility factor is defined as the ratio of usable performance to inherentperformance that is attainable by service pilots in combat conditions.Inherent performance is defined as the aircraft performance limit with“ideal” pilot compensation. Usable agility factor is averaged over theTurn to Target maneuver as well as Bob-up/Bob-down, Pirouette; Slalom,and Pull-up/Pushover maneuvers performed in both day and Degraded VisualEnvironment (DVE) conditions. For the Turn to Target maneuver, theusable agility metric is the ratio of the time required to complete theTurn to Target maneuver in an AH-64 Apache helicopter with “ideal” pilotcompensation, calculated as 4.1 seconds by the DoD, to the time it takesan actual pilot to complete the maneuver while maintaining at leastLevel 2 handling qualities. Note that Level 2 handling qualitiescorrespond to Cooper-Harper handling qualities ratings of 4 through 6.

The current fleet Apache AH-64A -flight control system allows servicepilots to exploit only 30% of the agility inherent in the Apacheairframe as indicated by a left-hand bar 96 in FIG. 9. Implementation ofa Partial Authority HACT Flight Control System (PAHFCS) that includesthe Heading Reference Command & Control Algorithm into a forcemodernized Apache helicopter is predicted to increase average usableagility factor by. 66%, allowing service pilots to exploit 50% of theagility inherent in the Apache airframe as indicated by a center bar 98in FIG. 9. Note that the PAHFCS utilizes the mechanical flight controlsystem implemented in current fleet Apache helicopters. Current fleetApache helicopters augment the pilot's direct mechanical control inputswith inputs from limited authority stability augmentation actuatorswhose control authority is limited to only roughly +/−10% of the pilot'stotal control authority. Thus the current fleet Apache flight controlactuator hardware is referred to as a “Partial Authority” flight controlaugmentation system because the actuators used for stability and controlaugmentation have only “part” of the authority available to the pilot.An Apache upgrade with the PAHFCS is a relatively low cost option forincreasing usable agility in the Apache, requiring extensive flightcontrol computer software changes but minimal flight control hardwarechanges. A partial authority HFCS upgrade including the HeadingReference Command & Control Algorithm is predicted to achieve the 2005RWV TDA Maneuverability/Agility Subarea Goal.

A more capable but mote expensive design option would be to replace themechanical flight control system currently used in the Apache with aFly-By-Wire flight control system that allows the HFCS to command fullactuator authority. The full authority HFCS upgrade would requireextensive flight control hardware changes and extensive flight controlcomputer processor and software changes. As shown by the right bar 100in FIG. 9, the full authority HFCS implementation would allow pilots toexploit roughly 65% of the agility inherent in the Apache airframe,achieving the year 2010 RWV TDA Agility/Maneuverability Subarea Goal.

The DoD defines connectivity between the “Increase in UsableAgility/Maneuverability Subarea Goal” and System Level Payoffs in theRWV TDA. For example, attaining the 2005 (Phase 2)Agility/Maneuverability Subarea Goal is predicted to increase missioncapability by 65%, reduce major accident rate by 10%, increaseprobability of survival by 4.5%, and increase mission reliability by 20%in comparison to current fleet Apache helicopters. Thus the HeadingReference Command & Control Algorithm is predicted to have significantbenefits in expanding mission capability, increasing flight safety,improving survivability, and increasing mission reliability.

In accordance with one aspect of the invention, a final headingprediction algorithm may be derived and based upon the solution to thecontinuous time domain control problem. A general derivation of aheading reference command and control algorithm is set forth below.

General Derivation of Heading Reference Command & Control Algorithm

There exists a state: X.

Its derivatives exist: {dot over (X)}, {umlaut over (X)},

, . . .

Limits on the derivatives are defined: {dot over (X)}_(MAX), {umlautover (X)}_(MAX), and

_(MAX) such that |{dot over (X)}|≦{dot over (X)}_(MAX) etc.

The basic relationships are defined as exponential: {dot over (X)}=−K X

Thus: {umlaut over (X)}=−K {dot over (X)},

=−K {umlaut over (X)}, {umlaut over (X)}=K² X,

=−K³ X etc.

Given an arbitrary initial condition, an optimal “at rest” condition maybe achieved as follows:

Initial condition defined as: {dot over (X)}={dot over (X)}₀, {umlautover (X)}={umlaut over (X)}₀,

=undefined

At rest” is defined as all derivative states=0.

There are two basic conditions that may exist.

-   -   1. The acceleration (deceleration) profile must be “captured”.    -   2. The acceleration profile is being tracked.        -   a. Constant (maximum) deceleration is active.        -   b. Exponential curve is being tracked.

Starting from an arbitrary initial condition the capture may be toeither the exponential or the constant (maximum) acceleration portionsof the profile.To the Exponential:$t_{1_{\exp}} = \frac{{1{K( {{\overset{.}{X}}_{0} + {{\overset{¨}{X}}_{0}t_{1_{\exp}}} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{1_{\exp}}^{2}}{2}}} )}} - {\overset{¨}{X}}_{0}}{{- {\overset{\dddot{}}{X}}_{MAX}}\quad{sign}\quad( {\overset{.}{X}}_{0} )}$${t_{1_{\exp}}^{2} - {\frac{2( {{K( {\overset{¨}{X}}_{0} )} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}} )}{K( {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )} )}t_{1_{\exp}}} - \frac{2( {{K( {\overset{.}{X}}_{0} )} + {\overset{¨}{X}}_{0}} )}{K( {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )} )}} = 0$$t_{1_{\exp}} = {\frac{{K( {\overset{¨}{X}}_{0} )} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}}{K( {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )} )} + \sqrt{\begin{matrix}{( \frac{{K( {\overset{¨}{X}}_{0} )} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}}{K( {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )} )} )^{2} +} \\\frac{2( {{K( {\overset{.}{X}}_{0} )} + {\overset{¨}{X}}_{0}} )}{K( {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}\quad}_{0} )} )}\end{matrix}}}$${\Delta{\overset{.}{X}}_{1\quad\exp}} = {{{\overset{¨}{X}}_{0}t_{1_{\exp}}} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{1_{\exp}}^{2}}{2}}}$${\Delta\quad X_{1\quad\exp}} = {{{\overset{¨}{X}}_{0}t_{1_{\exp}}} + {{\overset{¨}{X}}_{0}\frac{t_{1_{\exp}}^{2}}{2}} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{1_{\exp}}^{3}}{6}}}$To the Constant Acceleration:$t_{1_{MAX}} = \frac{{{- {\overset{¨}{X}}_{MAX}}\quad{sign}\quad( {\overset{.}{X}}_{0} )} - {\overset{¨}{X}}_{0}}{{- {\overset{\dddot{}}{X}}_{MAX}}{sign}\quad( {\overset{.}{X}}_{0} )}$${\Delta{\overset{.}{X}}_{1_{MAX}}} = {{{\overset{¨}{X}}_{0}t_{1_{MAX}}} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{1_{\exp}}^{2}}{2}}}$${\Delta\quad X_{1_{MAX}}} = {{{\overset{.}{X}}_{0}t_{1_{MAX}}} + {{\overset{¨}{X}}_{0}\frac{t_{1_{MAX}}^{2}}{2}} - {{\overset{\dddot{}}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{1_{MAX}}^{3}}{6}}}$If t₁ _(MAX) ≦t₁ _(exp) then: Δ  X₁ = Δ  X_(1_(MAX))$t_{2} = \frac{{\overset{.}{X}}_{0} + {\Delta{\overset{.}{X}}_{1_{MAX}}} - \frac{{\overset{¨}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}{K}}{{\overset{¨}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}$${\Delta{\overset{.}{X}}_{2}} = {{- {\overset{¨}{X}}_{MAX}}\quad{sign}\quad( {\overset{.}{X}}_{0} )t_{2}}$${\Delta\quad X_{2}} = {{( {{\overset{.}{X}}_{0} + {\Delta{\overset{.}{X}}_{1_{MAX}}}} )t_{2}} - {{\overset{¨}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )\frac{t_{2}^{2}}{2}}}$${\Delta\quad X_{3}} = \frac{{\overset{¨}{X}}_{MAX}\quad{sign}\quad( {\overset{.}{X}}_{0} )}{K^{2}}$Else:ΔX₁=ΔX₁ _(exp)ΔX₂=0${\Delta\quad X_{3}} = \frac{{\overset{.}{X}}_{0} + {\Delta{\overset{.}{X}}_{1_{\exp}}}}{K}$

The algorithm may be based on a set of equations that allow the “finalheading reference” to be computed continuously. Calculations may bebased on knowing the yaw bandwidth (K), yaw rate command {dot over(Ψ)}_(c) “derived from the stick”, maximum yaw rate {dot over(Ψ)}_(MAX), maximum yaw acceleration {umlaut over (Ψ)}_(MAX), andmaximum jerk rate {dot over ({umlaut over (Ψ)})}_(MAX) of the aircraft.Based on this aircraft information an algorithm was derived thatpredicts the heading response for a given yaw rate command. Thealgorithm is set forth below.${IF}\quad( {{{abs}( {\overset{.}{\Psi}}_{c} )} \leq \frac{{\overset{¨}{\Psi}}_{MAX}}{K_{1}}} )\quad{THEN}$$\quad{t_{1} = \frac{{{- K_{1}}*( {{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}^{\prime}}} )} - {\overset{¨}{\Psi}}_{C}}{{- {\overset{\dddot{}}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta{\overset{.}{\Psi}}_{1}^{\prime}} = {{{\overset{¨}{\Psi}}_{C}*t_{1}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{1} = {{{\overset{.}{\Psi}}_{C}*t_{1}} + {{\overset{¨}{\Psi}}_{C}*\frac{t_{1}^{2}}{2.0}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{3}}{6.0}}}}$$\quad{{\Delta\Psi} = {\frac{{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}^{\prime}}}{K_{1}} + {\Delta\Psi}_{1}}}$ELSE$\quad{t_{1} = \frac{{{- {\overset{¨}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}} - {\overset{¨}{\Psi}}_{C}}{{- {\overset{\dddot{}}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta{\overset{.}{\Psi}}_{1}} = {{{\overset{¨}{\Psi}}_{C}*t_{1}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{1} = {{{\overset{.}{\Psi}}_{C}*t_{1}} + {{\overset{¨}{\Psi}}_{C}*\frac{t_{1}^{2}}{2.0}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{3}}{6.0}}}}$$\quad{t_{2} = \frac{{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}} - \frac{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}{K_{1}}}{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta\Psi}_{2} = {{( {{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}}} )*t_{2}} - {{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{2}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{3} = \frac{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}{K_{1}^{2}}}$  ΔΨ = ΔΨ₁ + ΔΨ₂ + ΔΨ₃ ENDIF   Ψ_(PREDICTION) = Ψ_(C) + ΔΨ

Where,

-   Ψ_(c)=Yaw Command (Units =Radians)-   {dot over (Ψ)}_(c)=Yaw Rate Command (Units =Radians/sec)-   {umlaut over (Ψ)}_(c)=Commanded Yaw Acceleration (Units =Radians/sec    2)-   {umlaut over (Ψ)}_(MAX)=Yaw Acceleration Limit “Max Acceleration”    (Units =Radians/sec    2)-   {dot over ({umlaut over (Ψ)})}_(MAX)=Yaw Jerk Limit “Maximum Jerk”    (Units =Radians/sec    3)-   K =Yaw Desired Bandwidth (Units =Radians/sec)

Note the algorithm is broken in two parts, resulting in ΔΨ beingcalculated differently depending upon the state of$( {{{abs}( {\overset{.}{\Psi}}_{C} )} \leq \frac{{\overset{¨}{\Psi}}_{MAX}}{K}} ).$

Both PSI dot and double dot are signals generated within the Yaw AxisCommand model. They are labeled as Yaw Rate Command and Commanded YawAcceleration respectively. Both the Yaw Rate Command and Commanded YawAcceleration result from the pilot's directional input in lieu of limitssuch as Max Yaw Rate, Desired Yaw Bandwidth, etc. Note the Yaw Commandmay be derived in the Attitude Command Model as a Time Rate of Change ofAircraft Heading Angle Command, alternatively referred to as an InertialYaw Rate Command coming from the Yaw Axis Command Model. Sincehelicopters often hover at non-zero pitch and roll angles, it ispreferable to yaw about earth-fixed, or inertial, axes rather than yawabout the aircraft body axes for hover pointing tasks.

A variant of the foregoing algorithm has been found to perform well in afully-digital rotorcraft control system, and is set forth below:${IF}\quad( {{{abs}( {\overset{.}{\Psi}}_{c} )}<=\frac{{\overset{¨}{\Psi}}_{MAX}}{K}} )\quad{THEN}$$\quad{t_{1}^{\prime} = \frac{{{- K}*{\overset{.}{\Psi}}_{C}} - {\overset{¨}{\Psi}}_{C}}{{- {\overset{\dddot{}}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta{\overset{.}{\Psi}}_{1}^{\prime}} = {{{\overset{¨}{\Psi}}_{C}*t_{1}^{\prime}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{( t_{1}^{\prime} )^{2}}{2.0}}}}$$\quad{t_{1} = \frac{{{- K}*( {{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}^{\prime}}} )} - {\overset{¨}{\Psi}}_{C}}{{- {\overset{\dddot{}}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta{\overset{.}{\Psi}}_{1}} = {{{\overset{¨}{\Psi}}_{C}*t_{1}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{1} = {{{\overset{.}{\Psi}}_{C}*t_{1}} + {{\overset{¨}{\Psi}}_{C}*\frac{t_{1}^{2}}{2.0}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{3}}{6.0}}}}$$\quad{{\Delta\Psi} = {\frac{{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}}}{K} + {\Delta\Psi}_{1}}}$ELSE$\quad{t_{1} = \frac{{{- {\overset{¨}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}} - {\overset{¨}{\Psi}}_{C}}{{- {\overset{\dddot{}}{\Psi}}_{MAX}}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta{\overset{.}{\Psi}}_{1}} = {{{\overset{¨}{\Psi}}_{C}*t_{1}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{1} = {{{\overset{.}{\Psi}}_{C}*t_{1}} + {{\overset{¨}{\Psi}}_{C}*\frac{t_{1}^{2}}{2.0}} - {{\overset{\dddot{}}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1}^{3}}{6.0}}}}$$\quad{t_{2} = \frac{{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}} - \frac{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}{K}}{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}}$$\quad{{\Delta\Psi}_{2} = {{( {{\overset{.}{\Psi}}_{C} + {\Delta{\overset{.}{\Psi}}_{1}}} )*t_{2}} - {{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{2}^{2}}{2.0}}}}$$\quad{{\Delta\Psi}_{3} = \frac{{\overset{¨}{\Psi}}_{MAX}*{{SIGN}( {\overset{.}{\Psi}}_{C} )}}{K^{2}}}$  ΔΨ  =  ΔΨ₁  +  ΔΨ₂  +  ΔΨ₃ ENDIF   Ψ_(PREDICTION)  =  Ψ_(C)  +  ΔΨ

The variant of a heading prediction algorithm set forth above explicitlydefines and updates all possible states calculated by the headingprediction algorithm during each digital update cycle of the flightcontrol computer, ensuring constant computational throughput and timingrequirements for the heading prediction algorithm and eliminating anypossibility of computational overflow or underflow conditions resultingfrom ill conditioned inputs to the heading prediction algorithm. This isa robust heading prediction algorithm that eliminates the potential forcomputational singularities that may exist in previous alternativeheading prediction algorithms when those algorithms are used to predictthe terminal heading of aircraft that are compliant with ADS-33 handlingqualities specifications.

As still further alternatives, the following heading predictionimplementations have also been developed and are set forth below.

Exact Solution Implementation of Final Heading Reference Algorithm

To the Constant Acceleration:$t_{1_{CA}} = \frac{{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} + {\overset{¨}{\psi}}_{C}}{{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}$If  t_(1_(CA)) < 0  then  t_(1_(CA)) = 0${\Delta{\overset{.}{\psi}}_{1_{MAX}}} = {{{\overset{¨}{\psi}}_{C}t_{1_{CA}}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{CA}}^{2}}{2}}}$$t_{2} = {\frac{{\overset{.}{\psi}}_{C} + {\Delta{\overset{.}{\psi}}_{1_{MAX}}}}{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} - \frac{1}{K}}$If  t₂ > 0  Then:${\Delta\psi}_{1_{MAX}} = {{{\overset{.}{\psi}}_{C}t_{1_{CA}}} + {{\overset{¨}{\psi}}_{C}\frac{t_{1_{CA}}^{2}}{2}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{CA}}^{3}}{6}}}$${\Delta{\overset{.}{\psi}}_{2}} = {{- {\overset{¨}{\psi}}_{MAX}}{{sign}( {\overset{.}{\psi}}_{C} )}t_{2}}$${\Delta\psi}_{2} = {{( {{\overset{.}{\psi}}_{C} + {\Delta{\overset{.}{\psi}}_{1_{MAX}}}} )t_{2}} - {{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{2}^{2}}{2}}}$${\Delta\psi}_{3} = \frac{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}{K^{2}}$Δψ = Δψ_(1_(MAX)) + Δψ₂ + Δψ₃Else:To the Exponential:$t_{1_{\exp}} = {\frac{{K{\overset{¨}{\psi}}_{C}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}}{K{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} + \sqrt{\begin{matrix}{( \frac{{K{\overset{¨}{\psi}}_{C}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}}{K{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} )^{2} +} \\\frac{2( {{K{\overset{.}{\psi}}_{C}} + {\overset{¨}{\psi}}_{C}} )}{K{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}\end{matrix}}}$${\Delta{\overset{.}{\psi}}_{1_{\exp}}} = {{{\overset{¨}{\psi}}_{C}t_{1_{\exp}}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{\exp}}^{2}}{2}}}$${\Delta\psi}_{1_{\exp}} = {{{\overset{.}{\psi}}_{C}t_{1_{\exp}}} + {{\overset{¨}{\psi}}_{C}\frac{t_{1_{\exp}}^{2}}{2}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{\exp}}^{3}}{6}}}$${\Delta\psi}_{3} = \frac{{\overset{.}{\psi}}_{C}t_{1_{\exp}}}{K}$Δψ = Δψ_(1_(exp)) + Δψ₃ End  if ψ_(PREDICTION) = ψ_(C) + Δψ

Iterative Solution Implementation of Final Heading Reference Algorithm

To the Constant Acceleration:$t_{1_{CA}} = \frac{{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} + {\overset{¨}{\psi}}_{C}}{{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}$If  t_(1_(CA)) < 0  then  t_(1_(CA)) = 0${\Delta{\overset{.}{\psi}}_{1_{MAX}}} = {{{\overset{¨}{\psi}}_{C}t_{1_{CA}}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{CA}}^{2}}{2}}}$$t_{2} = {\frac{{\overset{.}{\psi}}_{C} + {\Delta{\overset{.}{\psi}}_{1_{MAX}}}}{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}} - \frac{1}{K}}$If  t₂ > 0  Then:${\Delta\psi}_{1_{MAX}} = {{{\overset{.}{\psi}}_{C}t_{1_{CA}}} + {{\overset{¨}{\psi}}_{C}\frac{t_{1_{CA}}^{2}}{2}} - {{\overset{\dddot{}}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{1_{CA}}^{3}}{6}}}$${\Delta{\overset{.}{\psi}}_{2}} = {{- {\overset{¨}{\psi}}_{MAX}}{{sign}( {\overset{.}{\psi}}_{C} )}t_{2}}$${\Delta\psi}_{2} = {{( {{\overset{.}{\psi}}_{C} + {\Delta{\overset{.}{\psi}}_{1_{MAX}}}} )t_{2}} - {{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}\frac{t_{2}^{2}}{2}}}$${\Delta\psi}_{3} = \frac{{\overset{¨}{\psi}}_{MAX}{{sign}( {\overset{.}{\psi}}_{C} )}}{K^{2}}$Δψ = Δψ_(1_(MAX)) + Δψ₂ + Δψ₃Else:To the Exponential:$t_{1_{\exp}} = \frac{{K*( {{\overset{.}{\Psi}}_{C} + {\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}}} )} + {\overset{..}{\Psi}}_{C}}{{\overset{...}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )}$${\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}} = {{{\overset{..}{\Psi}}_{C}*t_{1_{\exp}}} - {{\overset{...}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )*\frac{t_{1_{\exp}}^{2}}{2.0}}}$${\Delta\quad\Psi_{1}} = {{{\overset{.}{\Psi}}_{C}*t_{1_{\exp}}} + {{\overset{..}{\Psi}}_{C}*\frac{t_{1_{\exp}}^{2}}{2.0}} - {{\overset{...}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )*\frac{t_{1_{\exp}}^{2}}{6.0}}}$${\Delta\quad\psi_{3}} = \frac{{\overset{.}{\psi}}_{C} + {\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}}}{K}$Δ  ψ = Δψ₁ + Δψ₃ End  if   ψ_(PREDICTION) = ψ_(C) + Δψ

Approximate Solution Implementation of Final Heading Reference Algorithm

To the Constant Acceleration:$t_{1_{CA}} = \frac{{{\overset{..}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )} + {\overset{..}{\psi}}_{C}}{{\overset{...}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )}$If  t_(1_(CA)) < 0  then  t_(1_(CA)) = 0${\Delta\quad{\overset{.}{\psi}}_{1_{MAX}}} = {{{\overset{..}{\psi}}_{C}t_{1_{CA}}} - {{\overset{...}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )\frac{t_{1_{CA}}^{2}}{2}}}$$t_{2} = {\frac{{\overset{.}{\psi}}_{C} + {\Delta\quad{\overset{.}{\psi}}_{1_{MAX}}}}{{\overset{..}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )} - \frac{1}{K}}$If  t₂ > 0  Then:${\Delta\quad\psi_{1_{MAX}}} = {{{\overset{.}{\psi}}_{C}t_{1_{CA}}} + {{\overset{..}{\psi}}_{C}\frac{t_{1_{CA}}^{2}}{2}} - {{\overset{...}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )\frac{t_{1_{CA}}^{3}}{6}}}$${\Delta\quad{\overset{.}{\psi}}_{2}} = {{\overset{..}{- \psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )t_{2}}$${\Delta\quad\psi_{2}} = {{( {{\overset{.}{\psi}}_{C} + {\Delta\quad{\overset{.}{\psi}}_{1_{MAX}}}} )t_{2}} - {{\overset{..}{\psi}}_{MAX}{{sign}{\quad\quad}( {\overset{.}{\psi}}_{C} )}\frac{t_{2}^{2}}{2}}}$${\Delta\quad\psi_{3}} = \frac{{\overset{..}{\psi}}_{MAX}{sign}\quad( {\overset{.}{\psi}}_{C} )}{K^{2\quad}}$Δ  ψ = Δ  ψ_(1_(MAX)) + Δψ₂ + Δψ₃Else:To the Exponential:$t_{1_{\exp}}^{\prime} = \frac{{K*{\overset{.}{\Psi}}_{C}} + {\overset{..}{\Psi}}_{C}}{{\overset{...}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )}$${\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}} = {{{\overset{..}{\Psi}}_{C}*t_{1_{\exp}}^{\prime}} - {{\overset{...}{\Psi}}_{MAX}*{{SIGN}{\quad\quad}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1_{\exp}}^{\prime 2}}{2.0}}}$$t_{1_{\exp}} = \frac{{K*( {{\overset{.}{\Psi}}_{C} + {\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}}} )} + {\overset{..}{\Psi}}_{C}}{{\overset{...}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )}$${\Delta\quad{\overset{.}{\Psi}}_{1}} = {{{\overset{..}{\Psi}}_{C}*t_{1_{\exp}}} - {{\overset{..}{\Psi}}_{MAX}*{SIGN}\quad( {\overset{.}{\Psi}}_{C} )*\frac{t_{1_{\exp}}^{2}}{2.0}}}$${\Delta\quad\Psi_{1}} = {{{\overset{.}{\Psi}}_{C}*t_{1_{\exp}}} + {{\overset{..}{\Psi}}_{C}*\frac{t_{1_{\exp}}^{2}}{2.0}} - {{\overset{...}{\Psi}}_{MAX}*{{SIGN}{\quad\quad}( {\overset{.}{\Psi}}_{C} )}*\frac{t_{1_{\exp}}^{3}}{6.0}}}$${\Delta\quad\psi_{3}} = \frac{{\overset{.}{\psi}}_{C} + {\Delta\quad{\overset{.}{\Psi}}_{1}^{\prime}}}{K}$Δ  ψ = Δψ₁ + Δψ₃ End  if   ψ_(PREDICTION) = ψ_(C) + Δψ

Of the three implementations set forth above, the exact solution islikely to produce the most accurate heading prediction, but it is themost computationally demanding. The iterative solution is the more exactof the other two less computationally demanding solutions, but it maysometimes have convergence problems. The approximate solution will neverhave convergence problems, but it is the least exact solution.

The invention provides the pilot with a significantly reduced workloadand allows the pilot to focus more attention on other tasks, therebyimproving mission effectiveness and increasing safety. A systemconfigured in accordance with one aspect of the invention has beendemonstrated in an AH-64A helicopter simulator to achieve theaforementioned turn to target performance specification goals formaneuverability and agility of helicopters established by the U.S. Army.

Other aspects and features of the present invention can be obtained froma study of the drawings, the disclosure, and the appended claims. Forexample, although the foregoing has focused on piloted rotorcraft, theinvention could be used on other aircraft, machines such as robot armsand cranes, that need to be precisely pointed in a desired direction,and in connection with flight controls for remotely piloted vehicles,such as, for example, unmanned helicopters as means for a remote pilotand/or an autonomous flight control system to better control such avehicle. The invention is applicable to both fixed wing and helicopteraircraft in both “coordinated” and “uncoordinated” pointing (or hybrid)tasks. For instance, at a given speed, the heading rate (analogous toyaw rate for a hovering vehicle) is proportional to the bank angle in acoordinated turn. The heading acceleration is proportional to the bankangle rate. The heading “jerk” is proportional to the bank angleacceleration.

1. A computerized system that predicts a final attitude of a maneuveringaircraft, the system comprising: a computer processor; and an algorithmprogrammed into said computer processor and adapted to calculate, duringa maneuver, a predicted final aircraft attitude based on aircraftparameters wherein said algorithm includes calculations based upon modelfollowing control laws.
 2. The computerized system of claim 1, whereinthe algorithm further includes a plant canceller element.
 3. Thecomputerized system of claim 1, wherein the final attitude includes afinal heading.
 4. The computerized system of claim 1, wherein the finalattitude includes a final pitch attitude.
 5. The computerized system ofclaim 1, wherein the model following control laws are tailored tomaintain aircraft angular acceleration within specified limits.
 6. Thecomputerized system of claim 5, wherein the specified limits includelimits of angular acceleration about an aircraft yaw axis.
 7. Thecomputerized system of claim 1, wherein the model following control lawsare tailored to maintain aircraft angular jerk within specified limits.8. The computerized system of claim 7, wherein the specified limitsinclude limits of angular jerk about an aircraft yaw axis.
 9. Thecomputerized system of claim 1, wherein the predicted final aircraftattitude is calculated using an exact solution algorithm.
 10. Thecomputerized system of claim 1, wherein the predicted final aircraftattitude is calculated using an iterative solution algorithm.
 11. Thecomputerized system of claim 1, wherein the predicted final aircraftattitude is calculated using an approximate solution algorithm.
 12. Amethod of predicting a final attitude of a maneuvering aircraft, themethod comprising: providing a computer processor; programming analgorithm into said computer processor to calculate a predicted finalaircraft attitude based on aircraft parameters; and calculating, duringa maneuver, said predicted final aircraft attitude.
 13. The method ofclaim 12, wherein the algorithm further includes a plant cancellerelement.
 14. The method of claim 12, wherein the final attitude includesa final heading.
 15. The method of claim 12, wherein the final attitudeincludes a final pitch attitude.
 16. The method of claim 12, wherein thealgorithm includes model following control laws that are tailored tomaintain aircraft angular acceleration within specified limits.
 17. Themethod of claim 12, wherein calculating the predicted final aircraftattitude includes using an exact solution algorithm.
 18. The method ofclaim 12, wherein calculating the predicted final aircraft attitudeincludes using an iterative solution algorithm.
 19. The method of claim12, wherein calculating the predicted final aircraft attitude includesusing an approximate solution algorithm.